Question: Solve the equation e. 2x² − 3 = 5x
1. **State the problem:** Solve the quadratic equation $$2x^2 - 3 = 5x$$ for $$x$$.
2. **Rewrite the equation:** Move all terms to one side to set the equation equal to zero:
$$2x^2 - 5x - 3 = 0$$
3. **Identify coefficients:** Here, $$a = 2$$, $$b = -5$$, and $$c = -3$$.
4. **Use the quadratic formula:**
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
5. **Calculate the discriminant:**
$$\Delta = b^2 - 4ac = (-5)^2 - 4 \times 2 \times (-3) = 25 + 24 = 49$$
6. **Calculate the roots:**
$$x = \frac{-(-5) \pm \sqrt{49}}{2 \times 2} = \frac{5 \pm 7}{4}$$
7. **Find each solution:**
- For the plus sign:
$$x = \frac{5 + 7}{4} = \frac{12}{4} = 3$$
- For the minus sign:
$$x = \frac{5 - 7}{4} = \frac{-2}{4} = -\frac{1}{2}$$
8. **Final answer:**
$$x = 3 \quad \text{or} \quad x = -\frac{1}{2}$$